First, in both tales the power of the women in each story is stressed. Second, in Chaucer's "The Wife of Bath's Tale," the Wife of Bath is respected by Chaucer (the narrator and fellow-pilgrim): she has buried five husbands but has succeeded in being a resourceful and respected businesswoman. She is also a woman of faith who has made many pilgrimages, which is also worthy of mention, but has a strong personality as well—witty and funny, she offers "comic relief" to the rest of the pilgrims. It is not surprising that Chaucer (the author) has the Wife of Bath tell a tale that demonstrates the strength of women. (There are only three tales offered by women.)
The Wife of Bath is looking for a new husband. Understandably, her story promotes matrimony; but it also promotes the advantages of being able to look beyond the outward appearance to the "gem" that lies hidden within. (This argument applies specifically to her: she is a "wide" woman, with a "gap-toothed" smile, dressed in brilliant reds—the same color as her face. She might seem a little overwhelming at first.)
Her story is about a knight in King Arthur's court. One day, he sees a maid in the fields outside the castle gates and rapes her. He is brought before the King and the Queen. Arthur is ready to have him executed, but Guenevere asks Arthur to allow her to decide his punishment. Arthur allows it, and the Queen gives the knight one year to answer this question: "What is it that every woman wants?" Failure to find the answer will forfeit his life. So he travels a year. On the last day, an old hag promises to give him the answer in exchange for a wish. He agrees. The answer is that a woman wants her own way with men, in all things.
Saved from death, the knight must grant the hag a wish: she wants to marry him. He balks:
...the crone discusses true gentility and charity with the Knight.
Ashamed, he agrees and they wed, and then she turns into a gorgeous, young woman. She tells him she can be beautiful and unfaithful; or, faithful and ugly. He must choose. He wisely allows her to choose, and she becomes beautiful and faithful.
In Gawain's tale, at Christmas, a green knight enters the castle and gives anyone a chance to cut off his head; but in a year's time, the favor must be returned. Young Gawain thinks it will be an easy contest and takes a swing. The head comes off, but the knight picks it up, reminds Gawain to meet him in a year and leaves. The following winter, Gawain travels to the the Green Chapel. Arriving early, he sees a castle; the host invites him to stay for Christmas.
Gawain notices especially a beautiful woman, the wife of Bertilak...
So the lord hunts each day, while Gawain and Lady Bertilak stay home. Each day, she tests Gawain's honor, trying to seduce him, but he never strays from his code. The host and Gawain share what each received that day, and there is honesty between them. However, when the lady gives Gawain a magic belt to save his life, he hides its existence.
Ultimately, Gawain goes to meet the Green Knight, who spares his life for his honorable behavior—Gawain realizes it is really his host! He is embarrassed for hiding the belt, but Bertilak forgives him his wish to save his life.
In both stories, there is a knight who is tested—needing to be more honorable...this is accomplished in both stories by a wise woman (or two). Chaucer uses Guenevere and a hag with magical powers. With Gawain, it is Bertilak's wife.